Let *N* be the natural numbers 0, 1, 2, 3, .... Let [*n*] be the first *n* natural or hypernatural numbers (the hypernaturals are an extension of the naturals that include infinite "natural" numbers; the reciprocals of such infinite naturals are infinitesimal). Thus, [5] = {0,1,2,3,4}.

Suppose we want to model a fair lottery on *N* by using hyperreal infinitesimals. (What I say should extend to other constructions of infinitesimals.) Let the relevant probability be *P*_{1}. Thus, *P*_{1}({*n*})=*u* for some infinitesimal *u*>0. Various paradoxes follow: *P*_{1} is non-conglomerable (a corollary of this), this non-conglomerability can be made to spill out into other domains of investigation (see this) and we can't define expected values without violating domination (Paul Pedersen has proved this this summer).

But nevermind the paradoxes. I want to get at one thing that is fundamentally wrong about this approach: no matter which infinitesimal *u* we've picked, *u* is infinitely too small.

Here's why. If *u*>0 is infinitesimal, 1/*u* is an infinite hyperreal. Let *K* be an infinite hypernatural number such that *K*<<1/*u* (for instance, let *K* be the hypernatural closest to the square root of 1/*u*), where we say that *a*<<*b* for positive real or hyperreal numbers *a* and *b* if and only if *b*/*a* is infinite (i.e., *a* is infinitely smaller than *b*). Note that then *u*<<1/*K*. Now consider a second fair lottery, this time on the set [*K*]. It is clear that for the second lottery the probability of getting any particular outcome *P*_{2}({*n*}) for *n* in [*K*] should be 1/*K*, if we are dealing with hyperreal probabilities. None of the paradoxes follow if we do this [as long as we deal only in internal functions and internal partitions --*note added later*]: this assignment is both intuitive and stands up to scrutiny.

But now let *n* be any (finite) natural number. Then *P*_{1}({*n*})=*u*<<1/*K*=*P*_{2}({*n*}). But this is unacceptable, because *P*_{1} correctly represents a fair lottery on *N* and *N* is a proper subset of [*K*] (since *K* is infinite, we have *n*<*K* for all *n* in *N*). Thus, *u* is supposedly the individual outcome probability for a lottery on *N*, but *u* would be infinitely too small for the individual outcome probabilities in the case of a lottery on the (much) larger state space [*K*]. Thus, *a fortiori*, *u* is infinitely too small for representing the individual outcomes of a fair lottery on *N*.

And this is true for *any* infinitesimal *u*. So the lesson is that the individual outcome probabilities for a fair lottery on *N* must be bigger than every infinitesimal. But they must also be smaller than every positive real number, since otherwise they will add up to more than 1. So there is nothing for them to be. Such a lottery is, simply, not a probabilistically coherent concept.

## 2 comments:

A paper based on this idea is forthcoming in Synthese.

Preprint here.

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